s- ':J-0 OS 'bEJlHH EHHbi tf

HHCTHTYT RD.EPHbiX

. HCCnEJlOBAHHfi

E4-6630

V .G.Soloviev

STUDY OF HIGHLY EXCITED STATES

BY. MEANS OF A MODEL WITH ·MULTIPOLE

AND SPIN-MULTIPOLE FORCES . -

1972

~TIONS

t the Joint Institute for I be original publica­iNith Article 4 of the \ and co~munications t . published in future ::: collections. }

,''c:

1 " ~

:fposited publications ··.last fi11.ures of the

;i?h. · ·.!ch are distributed

I

~ Jer

nd ce

:.; i'

E4-6630

--------'

V .G.Soloviev

STUDY OF HIGHLY EXCITED STATES BY MEANS OF A MODEL WITH MULTIPOLE

AND SPIN-MULTIPOLE FORCES . .

OO<wnrmoz-mu!i C!!C'it"'W'f% 'i · en~;~~ oe~JJ~f-\o~x~.J

6~i5fWi~TL;('{A ' . • ~ ::nFrms ::w-~

!

l

.J l

In ref~Jl-4f:a new approach to studing· the structu-.,,

reof highly excited states has been .proposed. In the·fra ..

mework of this approach, the study of· highly excited sta- ·

tes proceeds along•two lines: general semi-microscopic.

description based on the operator form of the wave func­

tion of a highly_ excited s.tate and numerical· calculations

on the basis of the·models taking into account quasipar-. . . f /2,5/ . f . ·, .

ticle-phonon 1nteractions. In re s. · , in the rame-·

work of _the model taki{lg.into account. interactions lead;..

·ing to superconduct.i.ngpairing correlations and multipole

:..... forces, the complication of the· structure of nuclear sta-,

tes with. increasing excitation.energy has been studied

· and the. density of nonrotationa!' states in 239 'U has been

calculated'up to an excitation energy of 6 MeV. In the

present paper we suggest ·a more generai model describing · .

. interactions of quasipartlcles with phonons in which, .in .~. . t - . . . •

addition .to interactions leading to superconducting pair-

ing correlations, we take into account multipole and spin­

. mul tipole forces.

'3

~

1. The Hamiltonian of the Hodel

We formulate the model in the framework of the semi­

microscopic approach (ref./6/) for the case of a deformed

odd mass nucleus. A generalization to the case of a sphe­

rical nucleus will be. made elsewhere. The Hamiltonian des­

cribing interactions between nucleons in the nucleus is

written in the form

H = 8av +Hpair +H Q +Ha (1)

where Hav .. is the average nuclear field of the neutron

and proton systems 1 Hpair interactions .leading to super­

conducting pairing correlations, H0

mul tipole-mul tipole

interaction, Ha spin-multipole - spin-multipole interac­

tion.

We consider the interactions of quasi-particles with

phonons in a deformed nucleus with an odd number of nucle-

.ons161. In the odd nucleus the~e is one quasi-particle

in addition to quasi-particles·and phonons of the even­

even nucleus. The presence of this quasi~particle results

in a 'slight change of phonons. One usually neglects this

change and assumes that in an odd A nucleus phonons are . \

indetical with those in an even-even A .:.1 nucleus. The'

multipole and spin-multipole interac~ion constants are

fixed in defining phonons in even-even nuclei. Therefore

in the calculation of the quasi-particle-phonon interac­

tion there is not a single free parameter.

4

. '.~1 \

.. , I

I I -·

l· : !

?

The .simultaneous account of multipole

tipole interactions leads to· complicated se

for determining the phonon energy in.even-e

ref./6/. However; an admixtpre of spin-~ult . . .

effects little even low'-lyingquadrupole an

tes. Therefore 1 usually. one_..,.phonon states -. - ·---~----- - '

and with · K71

= . 0 1 . 1 , 2 , 3 ( K is the pro

angular momentum on the nuclear symmetry ax

cribed without introducing spin-multipole f 71 + other hand, the structure of the K-.=1 st

·1y affected/?/ by the spin-multipole intera

.the influence of multipole forces .is neglig . - . + . bf phonons with A=L. and K 71 =l ,.may turn o

tant.when explaining_ particular features of

from neutron resonances.

Our mathematical apparatus is c_apable

nons. Therefore we ~alculate '· A•?,..4 . phonons 1

majority of such states are not collective~

commit no error since when the root of the

tion for the phonon energy approaches the P•

function of the one-phonon ·state turns to

tion of the two....;quasi-particle state. For t:

definiteness phonons with A •-:::_.4 are calculab

pole forces.

Thus, we use in the model spin~spin in·

with A= 1 for describing K 71 =1 + s~ates anc

multipole interactions with A·~2 ·for describ:

with K71=0+, 2+, 3+; 4+, ••• , and with K77=1

5

---------:-~---:----,......-----------~--:- --------------:---:----c---;----~c-

'{ c----;c---~------

Hodel

amework of the semi­

he case of a. deformed

fhe case of a sphe~

The Hamiltonian des­

in the nucleus is

(1)

ld of the neutron

.s . leading to super­

multipole-multipole ·-·

n-multipole interac-

quasi-particles with

. odd number of nucle- ·

one quasi-particle

.onons of the even­

~asi~particie results

:tially neglects this

:ucleus phonons are

A.-1 nucleus. The

.ion constants are

~ nuclei. Therefore

cle-phonon interac­

teter.

The .simultaneous account of multipole and spin-mul­

tipole interactions leads t'o complicated secular equations

for determining the phonon energy in.even-even nuclei,

ref./G/: However; an admixture of spin-~ultipole·forces ieffects little even low'-lyingquadrupole and oc~upole sta­

+ + ) tes. Therefore, usua_~~. one..,.phonon states with K ~ =0 ·, 2

- -and with K17 = 0 , _1 , 2 , 3 ( K is the projection of the

.angular momentum on the nuclear symmetry axis) are des~:;·~ c • ·

cribed without introducing spin-multipole forces. _On the

other hand 1 the structure of the K." =1 + state is noticeab­

·1y affec.ted17 I by the spin-mul tipole · interaction 1 · while

:the influence of mul tipo1e forc-es is negligible. The role . - + bf phonons with 'A=l._ and K 17 =1 may turn out to be impor-

tant_ when explaining particular features of Ml transitions

from neutron resonances.

Our mathematical apparatus is c_apable of using pho . ..:

nons. Therefore we C:~lcu1ate ' A•?;..4 phonons, though the

majority of such states are not collective. However, we

commit no error since when the root of the secular equa:­

tion for the phonon energy approaches the pole the wave

function of the one-phonon ·state turns to the·wave func­

tion of the two~quasi-particle state. For the.sake of

definiteness phonons with A •>_.4 are calculated with multi­

pole forces.

Thus, we use in the model spin-spin interactions

with A= 1 for describing K 17 =1 + states and multipole­

multipo1e interactions with .\·~2 for describing. states • 1T + + + + 1T - - -w1th K =0 1 2 , 3 1 4 , ••• , and with K =0 , 1 1 2 , •••

5

,C.

An admixture of A·:::..4 phonons to phonons with A =1 1 2

and 3 makes the spin space to two quasi-particles wider.

Thus 1 if we take into account phonons with A =1 1 2,3, ••• 8

then the spin and particles of two quasi-particles ~un - - . + +

over all the values from 0 to 7 and from 0 to 8 • It

should be noted that in spherical nuclei, using multipole

interaction it is possible to describe only states with

I 77 =A (-1) >.. and to enlarge the spin space for two quasi-

particles.it is necessary to introduce spin-multipole

interactions w~th many A values.

Taking into account the secular equations for deter­

mining the phonon energy w 11 and ~·n (where by g we ·denote

Apj, .for multipole phonons, by n- we denote A=1,p

for spin phonons, j is the number of the root of the secu­

lar equation), .the appropriate part of the Hamiltonian.

(1) can be written in the form16 ' 7/

1 HM =H +H , vq a (2)

. q . ' (+) 2 Hvq= LE(q)B(·q,q)-..LL _1_ L (la(q,q)Uqq') (E(q)H(q'))

q 2 g y q,q' q . (E(q)+E(q'Jl..:.w2 g

+ Q gQg-

!_ L -1

-· L, vc-~1(1 11 (q,q' JB (q,q'J+l

11 (q,q')B (q,q'JJ(Q: +Q,. J+c.c.l,

.4 g '1/Yg q,q q,q " . .. .

(3)

6

.I l

I .j I

I !

B: =B!(p) +B! (n)

1 . . B (n)==- L L ..L. I

(a(l), ·u (-:. / (E( "S)H (S') ss ·ss#

---::---::---(€(S)+€(S'Jl -w 2

n u 2 n Yn ·ss'

+.!._I-· l_ I y(+J((u(t)~(SS')+ .4 n "1/Yn ·sa, ·ss' ·rrs,

·- +. ;;O> ~(SS!)) (Q n +I 'BS'

. Here Qs , Qn are the phonon ope~ators, £1

I

where c is the correlation function, A chel

E(~J single-particle energyr v~~ ==·uq·uq, t.

·u ( V = ·u v , + ·u , v where ·uq , qq q q - q q ·.

Bogolubov transformation coefficients. The

quantum numbers characterizing .the single-)

of the everage.field. is denoted by (sa) fo1

system and by ( qu) for both systems~ u..;.±J

elements of the Ap multipole moment opera1 . .·. . . /6 7/ .

spin operators are denoted as 1 .

ls(qt,q2) provided Kz±Jl=

Is ( ;.;; u ql,q2 ..

·-s ul (qt ,q2) . p:.;ovided Kt+K2=

7

---------:-------------:-~---~~

nons with A =1 1 2

!-particles wider.

with A =1 I 2 I 3 1 ••• a si-particles run ''· .. + +

from 0 to 8 . It

.ei 1 USing mUl tipole

:. only states· with

;pace for two quasi- . l :e spin-multipole

~~uations for.deter­

There by · g we ·denote

denote A= 1, ll

:he root of the secu­

:.the Hamiltonian.

q)+ f(q'))

~ '2 .~w g

(2)

(3)

i

l I j r I .I . I

i,

·' ~ 1

{ 4)

2 2 . Here Q g 1 Qn are the phonon ope~ a tors 1 ( ( g)=vC +(E(g)-A) J

where c is the correlation function, >.. chemical potential,

E(g) single-particle energyJ v~~ =·uq·uq' t. vq vq' ,

·u ('i;) =·uq vq' + ·u , v .where ·uq , Yq are the qq - q., q ·. Bogolubov transformation coefficients. The set of the

quantum numbers characterizing .the single-particle level

of the everage~field. is denoted by (su) for the neutron

system and by ( qu) for both systems; u=+l. The matrix

elements of the >..p multipole moment operators and the spin operators are denoted as/6 , 71

provided K 1 tJL = K2 ·

{6)

7

(5)

:t ,_

(1)(q q )= CICI 17 2

(1) -CICI

q 1'q2

(i(l) q1,q2

provided K1 ±1 = K2

Provided K + K = + 1 1 2 -

where

a(p~ =<q+la +(-fa :lq'+•> 1 -;CP} =<H:Ia ·+(-/a :lq'-·>7 ~ p ~ . ~ p ~

©'

a11

are the Pauli matrices. Further

B(q7q:)=; a;aaq'a7· +. . B( q 1 q ')=::£a aq-a a q 'a

. Cl

!B(q7q.')= l:aa+ a, 1 ~(q 7 q')=La+ a ; 1

a q-a q -a a q-a q a

a+ qa is the quasi-particles creation operator,

( f g ' . (+) 2 ( ) ( , ( q7 q ) U , ) W E ( q H q )) Cl qq g

y = ::£ g q,q'

[( E(Q)+ E(q')/ -w; )2

. . 2 (q(1)(s;s')U(-),) W (E(S)H(s')) a ·s;s n y = ::£

n ' ·s;s [(E(S)+E(s'Ji -w2 ] 2 . n

"-.·

2. Formulation of the Model

(7)

The wav€ function of a nucleus with odd number of

neutrons which describes states. with a given K" is

written as

8

' I j

I .. ,

I

I I

1JI (K"J=G 1 _!__:£{aT + :£Dgi a+Q+ +:£ Dni a+ Q + + i p - ·pa ·psu ·sa g psa ·sa n . · · ..j2 Cl gs n s

+ ::£ ~1g2

::£ FA1S2i a+ Q +Q+ + ::£ ·s psa ·sa· g

1 g2 n1'2

::£ F nln2i ·s ·psa

a+ o+ Q+ ·sa n

1 n2

+::£ _:£F~r;: ~+ Q+Qt +.1_. ::£ :£R2122A3i a+n+{ g,n s P sa g n y:r 212~3. ·s psa sue 2

1

1 nn2n3 i +-l:. l:R a+Q+ Q+

..j3""n '2'!1 ·s psa . sa n n2 Q+ + l: l: R222 n1 +

n3 2,~ris psa qsa

+ ::£. l: ·l'nnj. a+ Q+Q + Q+llJI 7 "nn ·s .PBcCI, ·sag n n 2 0 ..... 2 .

where 1JI0 is the wave function of the ground s

evet:t-even nucleus, by (pu) we denote the set o

numbers of the single-particle _sta~e with give

by !i the number of the state. The normalizatio

is

('Pi*(K")lJII (K"))~l=(CPI/11+L l: (D, 2 I /+1. l: (Dni / + 2 gsa psu 2 n;s,u psa

SS2·I 2 n,n I . · " I 2 + · · l: ( F j + l: ( F . 2 ) + ..!._ l: ( F .,n ) + l: l: 0

S,S2;s,U psU . n,n2 su psu 2FJ,g;s,a (!SCI . g,g2S3 ·sa

+ l: l: (R2S2ni 2 l:- l: R 2~n2i 2 l: l: Rn,n2,nj ·psu ) + ( · ·psu . ) + . . ( .;psa g,g2 n. ·sa S,n,n2 :sa n,n2 ,n3sa

9

K +K =+1 1 2 -

fl. ., , -) a . q -•>'

"il

a , q.a

ion operator,

iel . .

(7)

ith odd n~rnber of

given K" ·is

I J,-

1 I l

'

I

+

+

where W0 is the wave function of the ground state of an

even-even nucleus, by (pa) we denote the set of quantum

numbers ~f the single-particle sta~e.with given K" and

by !i the number of the state·. The normalization condition is

('P:*(K"JW (K"JJ~l=(Cp1 /11+L I (Dg

1 f+.L I (Dpniu / + i i · 2 gsa psa 2 n;s,CI s

g~ i 2 n n i i 2 gg2 e3 I 2 + I ( F . 2 j + I ( F ~ 2 ) + ..L I ( F gn ) + I I ('R ) +

" " . Cl ·psCI . Cl psCI 2 n,g;s,CI P. sCI . g,g2g3 ·sCI psCI ,.,,.2 ,s, . . n,n2 s

( 9).

l \.

.. We calculate the average value of RM ·=·H vq +'H ~

OVer State (8) 1 a.S·ca result We get.

. . . 2

('I'! (K17

)H '1'. (K"J=(C 1 /I c (p )+.I_};~ (c(s)+w,.. )(D g;u ) -I M I p . 2 g BU "' p B

(-) vps

L}; ~---= - 2 g su y Yg

. . . (+) .

fg(ps)Dgi +1._ L L(c(~)+w )(D ni /+l.._L L:l!.!_Jl>(J s)D01 +. u psu · 2 n ·sa n psu · 2 n ·say Yn u · P psu

. · . 2 nn2 i 2 +};}; (c(s)+w_. +w, )(F gl-2 1

) +}; I (c(s:J+w +w )(F. )+. g;g

2 su " .,2 psu "•'2 ·su· n n2 p sa

• gni 2 ·• g~g3i 2 + L L I (c(s)+w_. +w )(F ) + L L (c(s)+w +w +w )(R · ) +

2 g,n su "' n p s u g,g2;~ su · · g g2

g3

p s u

gg2 ni . 2 .. ·. ·. gnn2

i 2 + }; I (c (s)+ w +w +w . )(R ) + L L (c(s)+w +w +W )(Rp su )+

g,g2 ,{'. su .· g. g2 n · P, s u g,IJ•'2 su · g -n "2 . • : · .

+ I · ~ ( c( s)+ w n,n2· '"3 ·so n

"'2"3 i 2 + w +w )(R · )+

"2 "3 psu

(+) + }; }; ~ ( (1 )_ ni

-- -au , D , n,n2s,·s~a~v"Yn ss psa

-(1) ni nn 2 i + u , D , )F ·

ss ps -u psu

' 2

(-) v ss'.

-.I .I -=(f , yY . 11.11

2s,s,u . g

2

g2 g; . -g2 g; ' gg2 i . ·ts:s')D .. 8 .-af .(s·s')D:. )F +

• . p a . ps -a p s u

10.·.

, I

I I

I

'

I

,I d

'I .I i

(+)

+ ~ ~.n

v ;, .... ~ [ · ·ss · · · (I) .,1 -(I) 111 11ni _... -=(-au· , D , + u ,D , )F

·ss' ·.ly . ·ss psu ·ss -ps-u ·psu '.' 0 V n

(-) v , ss .

-yYI1

--,;3 ~

( f 11(·ss') D ni - f-11 r·ss') D ni )F ~n;] ps'-u ·ps'-ir ·psu

l . v 113" , 111121131 11 1111.

~ -= (f (ss)R , -uf s(ss)R : ' .I · ps u ps 11,112 ,113 ·s,su V Y

113

v (+)

+ -'3 .I }; ~ - (I) n_n2 "3 i -(I) nn2 n3i v --(uu; ,R, +u.,R,

"'"2'"3 s,s u V Y ss.. p s u· ·ss p·s .u

+ .I .I , 11.112 ,n ·s,s,u

"3

(+) V , 1111 nl I11L

[ ~ (- (I) .R 11112n1 - (I) R 2 F -:.! - au , +u , , ) yYn ss ps'a ss ps-u psu

-11 1111 nl lt:Jl - u f fss')R ~ ) F ] + .I ·

v (+) [~ •(1) l1t

. ·. p·B -u P 111 U 11,n,n , --(-au , R s,s ,u· . 1 Y. ·. ·ss P' .I

(-) . v , ~(f

yYI1

v "2

11 , . ~n 21 - 11 , 11nn2 i nn2 i . (ss) R . , - u· f (ss ) R , ) F ] I .

psu. ps -u p·su

11

n n -j ; 2

.sa

2 ) -

2 )+.

(+) v , .

+ ~ ~ [ . '88 . (1) -4. -4. -= (-u·u· ,

g,n ·8,8 :u V Yn '88

g; -(1) g; .. gnJ D , + u ,D , )F

p8 u ·s8 p8-u p8 u

(-) v , 88 .

g ni - g ni gn i (f (ss')D , -£ (ss')D , .)F ]

p8 -u p8 -u p8u

+ I. I.

. v (+)

(10)

-g gg2 ni 111ti - u f fss)R , ) F ] + I.

·. p·11 -u psu g,n,n ~ [ ·s8 ( •(1)R111tn,J -(1),RI11tn2 i Fgni -4. -= -uu· , ,. +u· , ) -

s,s:u yY. · ·ss psu· .·ss ps'-u psu "2

(-)

~'as' g . 111tn 21 - g gnn2 ; nn i - (f (ss') R . , - u f (ss') R , ) F 2 ] I.

y·Yg P s u ps -u p·su

ll

The energies of the nonrotational· states '1/. .and I

the functions C 1 D 1 F and "R ·are determined by means

of the variational principle

oi('P,*(K 17)HM'P1.(K

17))..:.'11;·[('P;,(K

17)'P

1 (K

17))-1]1= o. (11)

We performed a variation and a number of transformations.

As a result weohave the following system of equations

.. (v(-)/ (f:(ps)) 2 ( d( 1) (p"S))2 a 1 . ps

£(p)-'1/1 -4(j~ y(j f(s)+w(j-'1/i

1

4 n,s l:

(v(+)) ~ ps

y n

f(S)+wn- '11;

+

!.. l: 4

I__! l: f

(j 1

(j2 , (j · - (jz , a (pS) (ss) + a f-a ( ps) f css )

2 tj,tj2 ,j Yg Yg2 f(S)+wg-'1/;

rg2(ps)fg(ss')+afg(ps)ltss') (j(ji a a ]F ,2

f (S) + w - Tj. ps a {j2 I

!._ l: 2 n,n2

(+) (+) vps vsS'. --- X

v·Y~ yn2

1

+

X ( + 1 (1) (1) (1) - (1) nn2i

-----)(-a a (pS)a ,+a. (ps) a , )F ·. , + f(S) + wn- Tl; f(S)+w - 11 • a ss -a ss psa

n2 I

+ l: g,n

1 v c->v (+) -====-[ _ ps ·ss' (- ( 1) g . . _ ( 1) g >JY,. Y f(S) + w -., ua;s' f (ps) + a. , f (pS)) +

s n g ., i a ss -a

12

(+) (-) v v , ps ·ss (1) · .. g.· · (1) . - g .gn;_

(a (ps) f (ss') + au (ps) f (ss'))] F , a -a p·s a

+ f (S)+wn- '11;

. (.:-) (-)

gg i 1 1 l: F 2 =·- , (

f(s)+w +w,. -'1/i) psa 14 .ly y ·s-g

152 v g g2

v , v , ps · ·ss

f(s)+·wg-'1/;

. 1 1 g g -g2. ' f g ~ ·s') + - l: --=

(f 2 (ss')~ (ps)- al (ss) -a P 2 ~,·s2,s3 ,jYg ~ 2 3

g g . . -g . -g gg i [(I 2(ss )I 3(s ·s )+I 2(ss )I 3(s ·s ))F 3 +

2 3 2 2 3 2 p·s3

a

' R2 -g3 -g -g gg; + a·(l (ss 2 )f (s3 ·s2 )- t t·ss

2)i 3(s

3·s

2))F . 3 ] -

ps3-a

v(-) v(+)

v (-) v(-: ps ·ss

f(S )+ W 2 g

- _!_ l: 1 ·ss2 ·s3·s2 lh (1) -g2 -[-a(ll'S~)a,g .

8 +I (ss

2Jo: 2

n,·s2 ,s3 yY,. Yn E(S )+w - '1/. 152 2 (j I

3 2 s

gni ,3 1 .__..:_ __ g -(1) -g2 (I) )F ]+·- l: y (s )+w + (f 2(SB:J )a - f (s·s2) as ·s ps3 -a 14 g3;s2 g f 2 g

s 3 82 3 2 3

13

states TJ 1 . and

~rmined by means

· 1] I ;, o . (11)

)f transformations.

~m of equations

(+) 2 (1) .. 2 • ) ( d (p'S)) ~. __ a ___ _

Yn . f(S)+wn- TJ;

' -. 11;

'+) (+) 's · V:ssf. ---·x

1 y"2

(1)(·s\ -(1))F""2; + a p~a, ., -q ss p·s a

(pS)) +

! .

l

I I

(+) (-)

vps v·ss' (1) ~:· (1) ..,.~, ~ni. + --..,.----..,.-(a (pS) I (ss') + ua (ps) I (ss'))] F ,

u -u p·s a 0'

~2 ~ -~2 , f II , 1 ~ 1

v (_-~ v r-: ps · ·ss

v r-> vr->, pa ·ss

X

( f (ss') f (ps) -.a f (ss) (p'S) + 2 .w -a -a· 11. ·s ·s -'Y Y f(S )+w -TJ

-;,• 2' 3 v II g 2 II i 2 3

g g . -g -~ gg i [(f 2 (ss )I 3(s ·s )+ f 2(ss )I 3(s ·s )).F 3 +

2 3 2 . 2 3 2 p·s3

a

R -II - g -·g gg i + u·( I 2

(ss ) I 3(s ·s ) - I t·ss ) I 3(s ·s )) F 3 ] -2 3 2 . 2 3 2 p

83-a

13

(12)

X

•

i !

'

X }; (-) (-) g g3 -g - g gg i v v [ (I 3(-ss ) f. (s ·s ) + I 3 (ss ) I 3 (s s ) F 2

s3 ss s s 2 3 2 2 . 3 2 ·p·s a

2 . 3 2 3

. g - g - g gg i +a(l 3(ss

2)1 3(s

3·s

2)-l 3(ss

2)1g3(s

3s 2 ))F 2 ] -

p8 -a 3 .

v (+)

...!_}; ~. X

4 n, s2 ..;y;- c(s2 ) + wg +wg2+wn- TJ i

l

· 1 1 g · - - g gni xI-=- I. v(-) [-a(a() I 2(s s )+a:(I)I 2 (s s )) F

.1y s s s ss2 3 2 ss 2 3 2 ps a v g2 3 3 2 3

(I) g (I)- g gni +(a 1·

2(s s )-a 1 2cs s J) F ]

·ss I 2 882 3 2 p 8 -a· 2 3.

f1 I. v (+) [(a(I)a(1) + ~(I);;(I))F gg/

ss s s ss ·s -s s 2 3 2 2 3 2 p 3a .JYn .83 s 8

3 2

OJ- (1) -<,-(I) (1)) F g~; ]I a (a a -a a ,

·8 82

s3

82

8s2

8 3

82

p 83 -a.

We should also perform a symmetrization in

the r.h.s. of (13),

14

q

+

and

+

(13)

q2 J.n

'\' "f:

\ ,('

v )

l l ! t i 't

} [

r

.Jf~. I r

'

,,~

. 1

I I I· \·.

'1:

( ·,!)

.}

I. ~

~

,, ! I'

'I \

i ~!

~~ ,( il J

nn j (c(s) + w + w - TJ ) F · 2

n n2 i p·8a·

. 1 1 ~-I.

,4 .JY Y 8, n n

2

1 1 +- I. --2 n•s;s .JYY

3 2 3 n n 2 3

- (1) (I) , -(1) (I) (+) (+) aas.8'.0: (p8 )+ 0: ,a (pi<'

v , v . , a ss -a p 8 ·8s ,

. c(s ) + wn - TJ i

vMvM m ss2 ·s3·s2 (1) (1) - (1)- (1) --[(a· a +a a )F

ss ·s ·s ss ·s ·s ·s c(s )+w :- TJ· 2 3 2 2 3 2 P.:

2 n 1

-a (1)";; (I) ) F nn3 i ] - _..!_ S82 s3s2 P s -a 2

I. 1 v(+)v(-~

·8s2 '8f82 [-a(

3 g,82,83 \1 Y Y · c (s )+w -TJ g n:z 2

- (1) - g gni (I) -g - (1) g , +a . I (s ·s )) F + (-a I (s s )+a I (s s ))

83 s 2 3. 2 p"3 a ·s82 3 2 s82

3 2

.3 I. +-

.4 n 3 g2 g3

1

y n3

v (+) v (+)

S82 si"3 ---------------------x c (s ) + w +w + w - TJ.

2 n n 2 n 3 z

[( (1) (1) - (1) - (1) ) F nn2

i (- (1) x a a +a a +a a ·882 s

382

8s2

s3

s2

p·s3

a ·ss2

15.

(I) (1 a -a

·s ·s ·ss 3 2

(13) ,,-.~-\ .

' II :..:,

- ' a.

:ion in q and ~n

+.l. ~ 1

, ·s

2 n>s,s yYY 3 2 3 ·n n

2 3

( 1)- ( 1 ) nn i 1 -a a )F 3 ]--

2 ss2

s 3

s2

p s3

-a

v (+) v (+)

.3 1 ·ss2 si"3 + - ~ --.4 y

£ (s2) +w +w n31l2 113 n3 n n2

+

. . ( 1) ll [-a(a f(s·s)+

ss· 3 2 2

X

+ w - TJ n 3 i

15.

'< ·-.:~

v (-) (-) 1

~ 1 'SS2 Vs3·s2 ~ ~

+- I [ f (ss ) f (s ·s ) -14 + ~-s ·s yY £(s2)+ w~ +w +w -11. - 2 . 3 2

• 2. 3 g n n2 I yY~

- ~ - ~ nn.. i ~ - ~ - ~ _ nn 2 i + f (ss2 )f (s ·s2 )) F 4 . + a(f (ss )f (s ·s .)-1 (ss )f (s ·s ))F ]-3 ps a. 2 3 2 2 3 2 ps

3- a

3

v (+) s3·s2

yYn

g ( 1) -~ - (1) [-a(f (ss2 )a +f (SS )a .

·s s 2 ·s ·s 3 2 3 2

~ - ( 1) - ~ . ( 1) ~ni + ( f (ss2 ) a . - f (ss ) a ) F ] I,

8 3 8 2 2 ·s·s ps-a . 3 2 3

) F ~ni

p·s3a + (14)

and a symmetrization in n and n2 in the r.h.s. of (14).

gni (E(s) + w -f w -11. ) F

n g I p'S a

1 -1 --= 2 ./yy

V n ~

(-) (+) + v , v ,

·ss ps

( 1) , - ( 1) ~ , -a a , f (p·s) +a , f (ps)

ss · a ·ss a· ~ [ v (+) ·s' ·ss'

(-) v ,

pS £ (s')+ w~ -11;

f g(ss')i 1) (ps')- a f ~ (ss') i 1>( ps') a· -a

-----------]- ~ E (s') + w - 11 .

n 1

g rs ;s

1G

+

v (+) v (-) 1 ·ss2 s3s2 -·- X

yY Y E(s2 )+w,.-11; n ~2 .,

I ( 1) ~ , ;,· -~2 ' ' g~ i

f h ·s ) + ;; ( 1) x [-a (a f (s ·s )) F 2 + 3 2 ·ss 3 2 p·s

3 a ·ss2 2

- ( 1) ~ ( 1) ._ ~ . ~~ i + (a f is

3 ·s

2)- a f l's3 ·s2 )) F _ 2 ] +

ss2

. ss2 ps3

-a

1 + ~

v(+) v(+) ·ss2 ·sfs2

[ (1) (1)-,...(1)-(1) Fgn2 ;

(a a +a a )

n ·s ·s 2' 3

-JY y n n 2

c(s2)+i:u~ -11i ·ss

2 ·s

3·s

2 ·ss

2 ··s

3·s

2 p s

3a

( 1) - ( 1) ~n2 i 1 -a a )F ] + ~ -=

·ss2 ·s3s2 p·s -a ~ ·s ·s yY y 3 . 2! 2' 3 ~ ~2

(-) v (-)

vss2 s3s2 [(f~(ss2 c(s ) +w -11

2

-~ -~ ~ni ~ -~2 -~-+ f (Ss ) f 2 (s

3 s

2 )) F 2 + a ( f (ss2 ) f (s .·s ) - f (ss ) J

2 3 2 2 p·s a

-~ 1

v (-) v (+)

'SS2 'Sj"J ~ (1 -~ _ 1 · £:-a(f (ss )a ) + f (ss )a ()

E(S )+w - 11 2 s ·s 2 s 'I 2 3 2 3 n ,s ;s y Y Y

2. 2 3 ~ n2

~ - ( 1) -~ · . ( 1) · nn i + (I (ss ) a - f (ss )a ) F 2 ] +

2 s ·s 2 ·s s 3 2 3 2 p·s3 -a

v (-) v(-)

1 ~ 1 ·ss., I ·sis2 +- .w -- __.._ X 2 -. ' ~ .. s.-s -JY E(s2 )+w~+w~ +w -11 . • yY 2 2 3 ~2 2 n I ~2

1-7

~

~ (ss ) I (s ·s ) +

2 3 2

- ~ nn i -l(ss )I(S'S))F 2

]-2 3 2 ps3-a

+ (14) 7

r.h.s. of (14).

g , (ps)

1'

-+

v (+) v (-) . 1 ·ss 2 s3 s 2

_ __;:~~__;;_X

V~ ~2 f(S2 )+wg-11;

l I

I j

I

l l }

I l

( 1)

x [-a (a ·ss2

+ ~ n s ·s

2' 3

112 - ( 1) -g2 gg i I (s ·s ) + a I (S

3 ·s2

)) F 2 + 3 2 ss2 ps3a

1

yY y n "2

v (-) v (+)

v (-) v (-) ss

2 s

3s

2

E(s )+w -71 2

ss2 SjS:2 g 1 -g -----[-a(f (ss )a () +I (ss ;;;: ( 1) e(s )+w -11 2 s s 2 s s

1 -~ n ;s ;s y Y y 2 2 3 g "2 2 3 2 3 2

(-) v(-) 1 v 1

~ ----~ ·s3 s2 + 2 X g ,s s yY f (s2 ) +w

11 + w

11 + w - 71. yY 2 2' 3 g2 2 n 1

g2

1-7

nn i ) F 2 +

p·s3 a

l.t

I ' ,.

I !.

g s -s _,g2 Sni x [(f 2(ss )f 2(s ·s )+ f 2(ss )f (s ·s ))'F

2 3 2 2 3 2 p·83

a +

g -g - S g Sni + a ( f 2 (ss ) f 2 (S ·s ) - f 2 (ss ) f 2 (s ·s )) F ] -

2 3 2 2 • 3 2 p·8 -a -3

(+)

V8 "82 S ( 1) - g gg ; 3 [-a(f

2 (ss )a + f 2(ss J~J1> ) F 2 +

2 ·8 "8 2 v'Yn 3 2 8382 p·83a

S ( 1) -S2 ; ( 1) Sl!. i +(I 2(ss )a - f (ss )a J F -:1 ]I -

2 "8 ·8 2 ·8 "8 ·p·8 -u (15) 3 2 3 2 3

1 v (+) (--)

}; "83 82 1 v

2 -- ,. "8382 ( 1) s n '8 ,8 yY E (~ J +(i)s +CiJ +(i) -71

[-a( a f (s ·s ) +

2 2 3 "2. n "2 I v'Y s 882 3 2

_(1) -·s nnl (1) g (1) g nnl + a f (s s )) F 2 + (a f · (s ·s J -a f -(s ·s )) F 2 ] -

·s82 3 2 p·s3

a 882

. 3 2 ·s·82

3 2 p·s3 -a

v (+) .83 82

v'Yn2

[ (a ( 1) ·s8

2

a (1)

'83."2

-·(lJ _,(1) F Snl _(1) (1) (1)_(1) .. Snl]l +a· a ) +· o(a a -a a 1 )F •

8s2 ·s3 ·s2 ps3

a "8s2

·s3

82

·ss2

·s3s2

ps3-u

18

v(-) fg(·sl 1 . !li 1 p·s. a P ~

D =- . ps a · 2 v'~ r (sj' + CiJ - TJ :

+ .: };

(-) v.ss2

/J g I r(s)+w - TJ. yY ...

.g·· I g2 ,g2 "'2

.v (+) -g gg i 1 ·s82 ( 1) g

- a I 2 (ss J F 2 ) - }; (-:-au F . 2 p8-a ((S)+w-TJ.n,s2 v'Y

ss 2 . P' 2 . g I n

v(+) / 1)(ps) ni 1 p s. a

D =-- --- -----1

};

v (-) ss2 g

(f (s p8a 2 v'-Y c(S)+ w -TJ.

n n 1

+------{ (~) + wn -TJ i g,s2 JYg

- g .. gnt - a f (ss

2) F ) -

ps -a 2

v (+) ·ss

-__ 3_ • ( 1) g

(-au . F 1

}; ·ss:z . p c(s) +wn· -TJi n2,s2 v' Y,;.

. . '2

.• gg2g i R 3

p 8~1

,y.3 1 }; ; (-) ·(fSj(ss') F g~; -u·fg3(ss~) F g~

·s' .ss' - ps'a p·s

·R gg

2ni

2 v'Y · .. g3

=_!_ 1

r(s) + w +w· +w~ -TJ . g g2 . 5 3. I

v "(-} .}; I ·~ (fg2(ss)

2 c(S)+w +w +w -TJ. v'Ys psa g g2 :n 1

- Sni afg2(ss')F, )­

p8 -o

v (+)' ·ss'

. v'Y n

2

.. ( 1) gg2 i - ( 1) (-au F . +a F , , ,

·ss ps a ·8s p5

19

+

Sni ~ )) F ] -2 p·s -a

3

I ) F gg i . . 2 +

I -(15)

'(-) ·s s 2 (1) g ..:::...J:.L_[-a(a I (s ·s ) + ..jY ss 2 3 2

g

( 1) g nn t -a I -(s ·s ))F 2 ] _

· :s·s2

3 2 p·s -a 3

,.

I I I .l

. (- ( 1) (1) ( J) ( 1) · gnt ·~· \ l + u a a -a u ) F ] •

7 ·ss ·s s s 2 3 2 ·s 2 ·sJs2 ps3-o

)

v (-) ll f (p'S)

lli 1 ps a D =2 +

psa -JY~ ( (S) + W - TJ.

!l I

'1 ----.'~

(16)

1 v (+)

!lni - ( 1> !lni -g llll i ·ss2 (1) ~ F +a F ) ' -af 2 (ss )F 2) (~aa . 2

t (s) + wll - TJ i ss 2 .ps2-a

ps2 -a n,s 2 -JY . ss p s2

a 2

1

n

1 ~

l(S)+wn -TJi g,s2

(+) vss

1l !lni (! (ss ) F

2 ps a 2

-g . !lni - a f (ss ) F ) - -----

2 ps -a

3 . . ( 1) ---(-au F

2 ss:z

. ..j.3 1

~g2ni 1 v r->

=..1_ ·ss ,

( £ 112(~s') F !lni

·R -~ 2

, psa l (s) + w +w +w - TJ i V Yg

p·s a ll 112 n 2

(+) V.ss' --- . (1) gg2t - (1). F gg2t .I

(- aa · , F , +a· , ) , ·ss ps a ·ss ps'-a

- gn; a fs2(ss') F . , )-

ps -u ..jY

n

19

(17)

(18)

(.19)

! ! .

r

. gnn i · 1 1 v (-) . ·ss'· g nn I

l: I -- ( f (ss') F . 2 ..JY p·s'-u

R . 2 =- ---,----

p·sa 2 E(S) + cu +Ci.l +cu - TJ

1 g n n2

• .. . g

- nn i -a £,11 (ss') F 2

nn n · R 2.3'

p·s'-u

v (+) 1 ... ·ss' ( ) .,m

) - --===- (-au·· , F , .1y ·ss ps a V n 2

~ (+) . (1) F nn2

i ., ,v , (-oo· , , ·s'· ·ss ·ss . ps a

_ ( 1) ~Jnl ·1 +a , F , ) ,

·ss ps -u

+a(l) Fnn2i ·ss' · ps'-u

p·sa

..;73 1 =-y·-=·

-../Yn 3

f (S)+ Ci.1 + W + Ci.l - TJ1 n n2 n_

3

(20)

(21)

The r.h.s. of expressi~ns (18-21) should be symmetrized

in g, g2 ,g3 ; g, g 2 ; n,n2

and n,n2

,n3

, respectively.

If we find the functions F from eqs. (13) , (14) and

(15) and insert them in (12) we get a secular equation

for determining the energies of nonrotational states TJ;

Knowing TJ. we obtain the functions D and R from eqs. (16-21) I . . .

and the function c from the normalization condition (9).

If we reject the spiri phonons Qn we are led to the system

of equations which was derived in ref. 151

3. An Approximate Solution

Eqs. (13), (14) and (15) contain coherent terms with . . f g , 2 d , ( 1) ( ',l 2 squared matr~x elements, ~.e. ( (s,·s)) an 1a · ss)J . and

· a a . noncoherent terms with the products ·

f 11(ss") f 11

,(s's"), ·a (1)(ss')u( 1) (s.'s") a· a a a f 11 (ss") ~ ( lJ(s's") •

a a

20'

..

We keep t~e noncoherent free. terms and·.reject

herent terms containing F. As a result we get

for F in the explicit form·

F gg2;

p·sa

. 1 1

i i(-) f (s) + Ci.l +·cu .;.. TJ. - S (s)- T ., (s).

g g2 I gg2 gg2 8-../Y Y

g g2

g , g ·, . _g , g ~. , • f 2(ss) f (ps)-af 2(ss)f (ps) .

a a X l;

(-) . (-) v v . [

ps' 'ss ,. +

f(s)+Ci.lg -.,.,; , ·s

g g . g2 . f 2(SS') f 2 (pSJ- a f g(ss') f ( p s')

+

F nn2 i

p·sa

X 2 , s

gni F

psa

a -u . ] ,

£(s') + Ci.lg -.,.,; 2

1 1 = . i T i(+) (

8 ...; Y y £ (S) + Ci.1 + Ci.1 - T/. ~ S . (s) -. n n2 · n n 2 . 1 ·. _nn 2

nn · 2:

v(+)v(+)[ 1 +-----

1---] (-uu(1)j 1>(p·s~

ps'·ss' £(s')+Ci.l - 11 . · £(S')+Ci.l - 11 . ·ss' a n 1 n 2 . 1

-1 1

- 5 j (S) - -yi(-) (S)-gn gn "2-.JY y g n

£ (s) + Ci.l_g + Ci.l n - 11;

21

•

g nn i 'I (ss')F 2

p·s'-u

( 1) gni . , F , ) I ,

s ps -u _

(20)

(21)

ld be symmetrized

1 respectively.

eqs. (13) 1 (14) and

secular equation

ational states ~~

nd R from ·eqs. (16-21)

ion condition (9).

led to the system 15/

:o~erent terms with md · (u( l) (ss')) 2 and

(J

( - , ·. (1) ' _(ss )u (sfs")• .. (J

t ! .

( I I J t

:1 I

·I ·' l

l !

~

We keep t~e noncoherent free terms and:.reject the· nonce- ·

.herent terms containing F. As a result we get expressions

for F in the explicit form·

X :£

+

, ·s

X :£ , B

gni F

- 1

1

-1

"2-JY y g n

1

(22)

] ,

1 X

(23)

21

- ( 1> f ~; ·s') + _,( 1> f ~ r' ·s'_) uu , tP u , tP x ~ I v(+) v (-) ·ss- u 88 -~q + v(+) v(+) x , , , ,

·s': ·ss · ps r(s)+cu~ _711

·ss ps

f ~ ') ( 1) , 1 ~ , ( 1) , (ss u (ps )-u (ss )u (ps)

u -u X I.

t ~s) + cu n -711

The following notation is used

s ~~ 2

(f~2(ss')) 2 . 1 ~) 2 . . .

(B) :: _, ~ (V ) ( U + 14yYY ·ss' t(B)+cu -,

g ~2 . g I

(f ~(BB)) 2

u :] ,

E (B)+cu~ - 71 1 2

(24)

(25)

I s (B) :: 1 (+) 2 1 1 ( 1) 2 -- ~ (v ) [ _+ ](u (BB')),

(25 ')

nn2 -- , u

.4y Y Y ·s' BS t(B') +cu -:71 t(B)-cu -, n n 2 n I · n 2 . 1

I 1 (u(1>(ss))2 ~ . 2 S (s) :: - ~ (v (+) / u 1 (-) 2 ( f (ss))

Sn Yn ·s' · ·ss' · , . + - ~ (V ,) __ u __ _

·r'c-> (.sJ

~~2

i! ~ ~ .4 ~ .••.

:3

T i(+) 1 (B):: - ~

14 n ~

~~2 ·s ,

E(B)+cu -, Y .. ·s' ·ss g I •

(~~7)2 y ~3

(V (+! )2 ·ss

y n

' . 2 ( f '3(as)) u

E(B')+ CUg +·cu~ +cug - 711 2 3

( u (1) (as) ) 2 U·

t(B')+cu +cu -to -71. . ~ ~2 n 1

22

,

£ (B) + cun - 711

<is' l>

.(26) I

{26')

T i(+) .3 . ( v (+))

2 (1) _ ~- ~ ss' (u u (ss'))

2

4 , . y -n3 ·s . n

. 3

(s) nn 2 c(s')+cu +cu +w -TJ.

n n 2 n3 '

(-) 2 g (ss')) 2

(I I(-) 1

(v , ) u

~ ss

1 (S) = - ~ nn

2 4 g " ,

yg c(s')+w +w +wg- 7J. n n2 '

(-) 2 g2 , 2

i (-) 1 (V , ) (f (ss))

ss a 1 (s) = - ~ }; ----

gn2 2 g2 "

, Yg E (s'J + w + (J) + w - 7J

2 g g n i 2

,' (a (

1) (ss'))

2 (+) 2

i (+) 1 (V ·,)

1 (s) }; }; ss. a =-

gn 2 , y ( (S ') + W + f,J + l>J - 7] n2 s n2 g n n2'

The functions F def1ned by (22) 1 (23) a

ins~rted in (12i • Then we obtain the explic~t

the secular equation for determining the ener

'nonrotational states. Now it is not difficult

te the functions c 1 D 1 F and R and thereby

expressions for the wave function (8) .

An approximate solution of the secular E

must not strongly differ from the exact one

role of the rejected noncoherent terms is no1

wever, this le~ds to the appearance of extraJ

To exclude these roots it is necessary to tal

count a part of noncoherent terms in eqs.(l3;

(15) •

' 23

1- v(+) v(+) x ·_ss' ps'

(24)

(25)

1 .. (1) 2 ---_..;.-](u (ss')), €(B)-ru -'1 u

. "2. i' (25')

g ' 2 (I (SB'))

(-) 2 u ~ (v ,) -------

'1

Yg ·s', ·ss

·ru - '1 'IZJ . i

-t.l -~ 1

n i

r (B)+ ru - 71 n i

(25' :',

. (26)

(26')

) 'i

. }

:I

I :1 ,,

( v (+}/ ( 1) , 2 i(+) .3

(a (SS ))

T ~- ~ ss a

(S) --- -----""2 4 ·s

, Yn c(s')+w +w +w -TJ. "3 3 n n 2 n3 1

(27)

(V(-;) 2 11

i (-) 1 ( f (ss'J) 2

T (S) ~ ~ ss a .

nn 2 4 11 .. , yl1 c (s') + w + w + w - i] . . n n 2 11 1

, ( 27 I)

(V (-;) 2 112 2 i (-) 1

( f (ss'JJ

T ~ ~ ss a

(s) - ----11n 2 2

112 yl1 E (s'j + w + U) +w .. - TJ. 2 11 11 n I

(28)

2

(+) 2 (1) 2 i (+) 1

(v ·,) (a (SS'))

T (s) ~ ~ ss. a

gn 2 y £ (S ') + W +(I) + U) - T} "2 ..

"2 11 n n 2

(28 I)

The functions F defJ.ned by (22) 1 (23) and (24) are

inserted in (12). Then we obtain the explicit form of

the secular equation for determining the energies TJ; of

'nonrotational states. Now it is not difficult to calcula­

te the functions c 1 D 1 F and R and thereby derive the

expressions for the wave function (8) .

An approximate solution of the secular equation

must not strongly differ from the exact one since the

role of the rejected noncoherent terms is not great. Ho­

wever, this leads to the appearance of extraneous roots.

To exclude these roots it is necessary to.take into ac­

count a part of noncoherent terms in eqs.(l3) 1 (14) and

( 15) •

23

i. '

i.

\· I, l·

.I

As a first step toward studying this model we

should investigate a particular case when in the wave

function (8} we put R = o • Then the equations read

£ ( p) - TJ. - ..l. ~ ~ 1 .4 g ·sa

v (+)

v (--) p·s

...jYg

g gi I (ps)D + a p·sa

1 " " p·s (1) r ·s D n i ;t - ~ ~ -- a ,p )

4 n ·sa yY a p·sa 0'

n

v (-) 1

(29}

1 (-) gi ( £ (s) + w - TJ. ) D - p·s a 1

I (ps)-- ~ I~ - v g 1 p·sa 2 -- a 2 y~ ·ss

, yY ·s s g 2 g 2 3 2

g g - -11 . g. (I 2(ss ) I 2(s ·s) + 1 112(ss ) I 2(s ·s ))D

1

2 3 2 2 3 2 p·s3

a X ··-+---··· -------- .

£ (s ) + w + w -·TJ 2 g 112 l

.. g -g2 -g g . . a (I 2 css ) I (S ·s ) -I 2 (ss ) I 2 (s s )) nl1'

. . 2 3 2 2 ___ 3 2 p~a + +

£ (s2 J +·w;+wg2 -~ 1

v (+) v (+) . 1· ·ss2 s3 s2

Y £ (s )+w +w,. -TJ. n 2 n 5 1

[(a (1) a(1) +;;.(1);; (1) )'D g; +· ss ·s ·s 'ss ·s ·s p·s -a

,2 3 2 2 3 2 3

+~ n

+a (u ( l) ·ss2

. gi c 1) c I> _ c 1J J ·n 1 a· -a a .+

·s 3

·s 2

·ss 2

·s 3

·s2

p·s3 -a

24

(-) v

·s ·s· 3 2

X

c+> . c-> v v

+ ~ 1 ·ss2 ·s3 ·s2 ( 1) g - (1)- g

------[a( a f (s3

s2

) +a f (s n

...JY Y £(S )+w +w -·TJ. ss2 . ·ss2.

gnc 2 n g 1

~

(1)-g -(J)g •. ni + (a f (s s ) - a · I (s ·s )) D ] I

·ss 3 2 ·ss 3 2 p·s· -a 2 2 3

v (+) n i

(£(S) + w - TJ. )D n 1 psu

1 +-

2 p·s . CT (1) (p'S)

1 - ~ 2 ·s s

2 3

- ( 1) + u (u

ss 2

CT

1.~ n

( 1)

·s ·s 3 2

yY n

(+) (+) 1 v·ss

--v

2 s3·s 2

y £(S )+w +w -TJ. "2 2 .n n 2 '

( 1) _ ( 1)) D n i ] + CT CT

·ss 2

·s3

s 2

p·s3 -u

v (-). (-)

CT

( ( CT ( 1)

·ss2

o,

(I? a +u

s s 3 2

+ ~ g

1

~

·ss2 v s ·s 3 2 g

[ ( f (ss ) f g - g -£ (S2)+w +w 'I . 2 (S 3'S2) + f (SS ) f

g n i 2

g -g -g g ni +a ( f (ss

2) I (s3 ·s2 )- f (ss2 ) I (s

3·s

2 )) Dp·s _) +

. 3

25

.ng this model we

.se when in the wave

e equations read

s ·s 2 3

g;

·s3 a . ,

(29)

1 (-) (-) v V X ,

Y ~ ·ss 2

-··-+---·--· ·-··--·· .

:I !

i'

+ ~ n

1

2

(+) (-)

1 v v.s ·s 882 3 2 ( 1) g - (1) -IJ n i

___;--~--- [a(a88

f (s3

s ) + a f (s s )) D + 2 2 . ss

2 3 2 p·s

3 a 'lfY Y £(S )+w +w -·TJ.

1J n • 2 n g 1

~ ~ 1

·s s n y 2 3 n2

1 +-

2

v (+) p·s

--===-·a (1) (p'S) 'lfY a

n

(+) (+) v·ss v s ·s

( 1) 2 3 2 [(a £(S )+w +w -TJ. ·ss

2 2 n n2

1

o. (30)

(1~ - ( 1) +a a

s s ·ss 3 2 2

_ ( 1) n i a ) D + ·s s p·s3 a

3 2

_ ( 1) ( 1) . ( 1) _ ( 1) D n i + a (a a -. a a · ) · ss ·s ·s ·ss ·s s . _ ] +

+ ~ g

1

~

2 3 2 2 3 2 P 83 a

f-"

I I

I I I

I I· l

v (:""") v (-) ·ss2 "8 3 8 2 S (1) -g - (1) Eli 1

+ ~ n

.. [a (I (ss )a ·+I (ss ) a )"D + .ry.y ,E(s

2)+cu,.+cu.-7J. 2_·s3·s2 2 ·s3 ·s2 p·sa ·

V n g · s n. 1

- g ( 1) g - (1) g i + ( f (ss2 )a - f (ss ) a ) D ] I

·F s~2; p·sa

F nn2 i

p·sa

s3·s2 .2 ·s3s2. p·s3-a• o,

1

(-) ~ . - ~~ ~ v , [ f (ss

2) D ~ 1 , -ufJS(ss) D ,

s' s·s · p·s a ps -u

2..; y~2 E (S) + CUg+ CUg2 :- TJ;

1

(+) (1) . nl _ (1) ni ~ v [a a , D , - a , D ,

·s' s·s' · ·ss 2 p·s a ·ss p·s -a

2VYn2

f (S) + CU + CU - TJ i

(31)

(32)

(33)

~nl v (+) ' .

I I!

,-~ i

1 1 . ~ I = 2 i(s)+cun +cu41 - ~~ s'

ss' (1) 411 - (1) ~~ ---: (au ,D. , -a , D , )+

V y ·s ·• ps a· ·ss p·s -'u F

psa

n

V' (-) . ·ss' S n·i . -g ni _ · (f (ss') D , ,-a.l'(ss') D , ) I,

.1y · pscx· .: p·s -u v g .. + (34)

If in eqs.\ (30) and (31) ~e reject the noncoherent

terms, obtain the ·functions~· D · and · inse~t· · th.erii in eq ~/ ·

(29) the secular equation takes on the following-form:

.<·*""~ .. , ..

E(p)-TJ

1• - ~ ~ .4 n s.

where

i(-) . 2 (s).

g

2 i (+) (s) g

.. .i (+) .2 . (S)

n

2 •·r-> rsJ n

.·· . 1s . 2

... (V (-;) )2 ( a (p"S}) 1 p

.4 ~ ~ i(-)

Y41

E(s) +cu41

-TJ1 -2 41 (s)-g . .•

v (+) (1) 2

(a (ps)) p·s a

= 0 2i(+) 2 1 (-) y E (s) + CU . -. TJ. - (S) - (S) n n 1 n n

! ~ ~ 14

g2 s'

1 ~ - ~ , 2 n ·s

1 ~ ~ -

.4 n ·s 2 .

~ ~ ~ 2 g ·s

,

g 2 (v (-))2 (f 2 (ss'n (1+ s .. g )

, . a 5 2 ~ y

g

(V (+) ··.2 ·ss')

y n

(+) (V·ss')

y "2

(v (7 / ·ss --y

g

E (S J + CU + CU " - TJ i . _s.. 5 2

(a (1) (ss')) 2 a

E (s') + cu + cu - TJ. . g. n 1

2 (1)' 2 .(a a (ss')) (1 +

. ,

Bnn

E (S') + CU + CU - !7J n n 2 l

(I g

(ss')) 2

E (S ') +. CU + CU - TJ . · g n 1

2

This model may serve. as a basis for st

structure of highly exc~ted nuclear states.

I am grateful to N.I.Pyatov and L.A.MaJ

ful discussions.·

' ' <~ ~:.

(31)

' ~ i ss) D , ps -a

(32)

) nl . D ,

p·s -u (33)

(1J ~~ - (1J • 'tn a a ,D , -a

1 D , )+ ·

·s·e. ps a· ·ss . p·s -·u·

(34)

ject the _nonconerent

ln~e~t: ~hell\ in ·eq ~ :he following form:

( (p) - .,.,

1' :£ :£

1 I :£ 4

~ . ..

v (+) p·s

.. . ,. . '

( 1) 2 . (u (ps))

u

.4 n s ( ··'+ - fi(+)(s'-2 1 (-)(s) = 0

( s, cun. -. T/ i n I n y n

where

1 :£ :£ 2 n ·s'

I(+) .2_ . (S)

n

f i(-J (S) = ....!. :£ I n 2 ~ .8

(V (-) )2 , Sl!l

y ~

(V (+), }2 ·ss

( (s) + cu + cu - T/ i ~ . ~2

( u (1) (ss')) 2 u

( (s') + cu + cu - T/ ~ n i

(1) 2 .( U U (SS ')) ( 1 + 8nn )

2

~ 2 ( f (ss'))

( (S ') + (U + (U - T/ · ~ n i

.)

(35)

(36)

(36')

(37)

(37 ')

This model may serve. as a basis for studying the

structure of highly exc~ted nuclear states.

I am grateful to N.I. Pyatov and L.A.Malov for use­

ful discussions.·

:

·i

j :1

. '

. . · References 1. B.r~··Conos&es. fl:~, ,12.,48 ,(1971).

2. B.r. ConOBbeB. Has. AH CCCP, cep cpua., ~ 666 ( 1971).

3. B.r. Conos&es. H<l>, · .!.§., 733 (1972).

4. B.r. ConoB&eB. ~tj"AH, ~ N9 4 .(1972).

5. B.r. ConoBbeB, n.A~ Manos~ ITpenpHHT OI1HI1, P4-6346, Jly6Ha, 1972.

6. B.r •. Conos&es. TeopuSI cnmKHbiX smep. HayKa, 1971.

7. A.A. Kynuea, H.I1. fiSITOB. H<l>, !!, 313 (1969).

Received by Publishing Department on 'July-27 r 1972.

..

28

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